= ~ Desingularization of non-dicritical holomorphic foliations and existence of separatrices

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Acta Math., 169 (1992), 1-103

Desingularization of non-dicritical holomorphic foliations and existence of separatrices

F. C A N O

Unioersidad Valladolid Valladolid, Spain

b y

and D. CERVEAU

Unioersit~ de Rennes I Rennes, France

Dedicated to Jean Martinet, in memoriam

Introduction

In this paper we complete the reduction of the singularities for non-dicritical holomorphic foliations of [5] and [7], in order to get only the so-called simple singularities. As a consequence, we prove Thom's conjecture about the existence of convergent separatrices, in dimension three. These results where announced in [8].

Let X be a non-singular analytic variety over C. A holomorphic singular foliation of codimension one over X is an integrable and inversible tTx, e-module of the cotangent sheaf f2x such that the quotient g2x/~ has no torsion. This means that each stalk ~e is generated by a differential 1-form

= ~

b i d x i ; bi6

~?x,e

i=1

such that if2 ^ dg2 =0 and g.c.d.(bi; i= 1 ... n) = 1. The singular locus Sing c~ is locally given by

Sing ~ = (bi = 0;i = 1 ... n).

It is a closed analytic subset of X of codimension 32. An irreducible element f 6 ~?x,e is a separatrix or an analytic solution i f f f divides Q ^ d f . This means that (f=0) is contained in a leaf, outside the singular locus. Analogously, a formal separatrix or a formal solution is an irreducible element f E ~x,e (=formal completion of tTx, e along its maximal ideal) such that f divides f~ A df.

The result in this paper concerning Thom's conjecture may be stated as follows:

1-928285 Acta Mathematica 169. Imprim6 le 20 aoQt 1992

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F. CANO AND D. CERVEAU

EXISTENCE OF SEPARATRIX THEOREM (dimension three). I f ~g is a germ o f holomorphic singular foliation o f codimension one over (C 3, 0) given by (if2=0) then one of the following two properties is satisfied:

(i) ~ has an analytic solution at the origin.

(ii) There is an analytic mapping a*: (C 2, 0)---)(C 3, 0) such that u*• is not identical- ly zero and the foliation given by (tr*Q=0) has infinitely many analytic solutions.

When (ii) holds we call the singularity a "dicritical singularity". In the two dimensional case, the existence of an analytic solution has been proved by Camacho and Sad [4]. In the three dimensional case, Jouanolou [15] gives a counterexample to the existence of a separatrix in the case (ii) above.

Like in the case of varieties (cf. [1], [13]), the reduction of the singularities intends to improve the singularities by blowing-up the ambient space X. More precisely, let Jr:X'-->X be the blowing-up of X with a non-singular center YcSing ~3. Then there is a unique singular foliation ~ ' over X' such that

~']x,_~-l(y) :

lx-Y.

We call ~d' the strict transform of ~ by zr. Note that, even if we blow-up repeatedly, we do not necessarily get that ~3' has no singular points. This can be easily seen by blowing-up ydx+xdy. Thus we can only hope to get "simple singularities", in order to have the following result:

DESINGULARIZATION THEOREM. Let ~ be a non dicritical holomorphic singular foliation over X=(C3,0). Then there is a sequence o f "permissible blowing-ups"

(1) X(1) <--X(2) <--... ^~r(1) ~r(2)

~<.N_mX(N)

such that the strict transform ~ ( N ) o f ~ under this sequence has only simple singulari- ties.

Let us explain somehow the above statements. First, let us recall the situation in the case dimX=2. Write

and put

if2 = adx+bdy, a(P) -- b(P) = O,

a a

D = - b + a - - ax ay "

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SEPARATRICES A N D DESINGULARIZATION OF FOLIATIONS

r~

P

" ~ Fa

Fig. 1

The point P is a simple singularity iff the linear part of D has two distinct eigenvalues a~:fl4=O and a/fl ~ Q+ (=strictly positive rational numbers). The simple singularities are persistent under blowing-up. In fact, the blowing-up of a simple singularity produces exactly two other ones, corresponding to the eigendirections. Moreover, there are exactly two formal separatrices Fa and F~ at P, which are both non-singular and tangent to the corresponding eigendirection. By Briot-Bouquet's Theorem, we know that Fa is always convergent. (See Figure 1.)

Now, we can choose a regular system of parameters (x,y) of ~x,e and ff~ which is written down in one of the following formal normal forms:

(i) f2=xy(dx/x+2dy/y); 2 6 C , ~,~Q_;

(ii) f2 =xyy s(dx/x + (e + 1/yS) dy/y); s >I 1, e 6 C;

(iii) Q=xy(xPyq) s (dx/x + (e + 1/(xPyq) s) (p dx/x +q dy/y); g.c.d.(p, q)= 1, s ~ > I ; (cf. Part II). There, we have that FatJF~=(xy=0).

Assume now that P is the only singular point of ~. Then, the two-dimensional desingularization due to Seidenberg [20] says that there is a finite sequence of blowing- ups at singular points

(,)

x(1) ,---x(2) , - - . . .

such that all the singularities in the last step are simple singularities. Let E(N) be the exceptional divisor produced by the sequence (*). The irreducible components of E(N) which are generically transversal to the strict transform ~(N) of ~ are called "dicritical components". Thus, the non-dicritical components are leaves of ~(N). Note that a dicritical component produces by blowing-down infinitely many separatrices at P. (See Figure 2.)

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dicritical component---~

Fig. 2

E(N)

We say that ~q is non-dicritical iff E(N) has no dicritical components. This is equivalent to say that ~d has only finitely many separatrices at P.

Let us restrict our attention to the non-dicritical case. Given a point Q fiE(N), denote by e--e(E(N), Q) the number of irreducible components of E(N) through Q. If e=2, then Fa U F~=E(N), locally at Q, and hence no other separatrix of C~(N) passes through Q. If e-- 1 and Q E Sing ~d, then either E(N)=Fa or E(N)=F~, locally at Q, hence there is exactly one separatrix FQ of ~(N) at Q with FQ~E(N). (See Figure 3.)

By blowing-down these FQ, we obtain a bijection

(formal separatrices of ~ at P ) ~ (points Q E E(N) N Sing @(N) with e = 1 }.

By [4], we know that there is always a point Q with e= 1 such that Ft2 corresponds to a nonzero eigenvalue, hence FQ is convergent and projects over a convergent separatrix F of ~ at P. (See Figure 4.)

)

N)

Fig. 3

E(N)

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E ( N )

,re.---

Q

SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS

Fig. 4

Thus, as mentioned above, Thom's question about the existence of a convergent separatrix has an affirmative answer in the case dimX--2.

Now, let us consider the case that dimX--n~>3. Let us fix a normal crossings divisor E of X. Here E plays the role of the exceptional divisor in an intermediary step of the desingularization process, hence in the initial step we shall put E=•. A dicritical component o f E is an irreducible component of E which is generically transversal to ~.

Consider a blowing-up :r: X'--~X with center Y. Note that if the center Y has normal crossings with E, then E'=:r-I(EU Y) is also a normal crossings divisor of X ' . We say that Y is a permissible center for ~ adapted to E iff, in addition Y satisfies a certain condition of equimultiplicity locally at each point (cf. [5], [7] and Part I). We say that c~

is non-dicritical iff E has no dicritical components and this remains true after any finite sequence of permissible blowing-ups (this definition is made relatively to E, actually, it deals with the initial singular foliation, before starting the desingularization process).

Roughly speaking, to say that c~ is dicritical means that for a certain non-degenerate two-dimensional section we can find infinitely many integral curves (cf. [6]). This corresponds to the condition (ii) of the Existence of Separatrix Theorem.

In opposition to the same phenomena in the two dimensional case, the dicriticalness is an obstruction to the existence of a convergent (even a formal) separatrix. In fact, the dicritical foliation given by the differential form

if2 = (xmy--z m+l) dx + ( y m z - x m+l) d y + ( z m x - y re+l) dz, m >t 2,

has no separatrices at the origin [15]. Thus, we may reformulate Thorn's question about the existence of separatrices as follows:

If ~ is non-dicritical, does (g have a convergent separatrix?

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/ . . / / .

e = l e = 2 e = 3

Fig. 5

Assume that @ is non-dicritical and that E is given locally at P by

U

^xi=O.

iEA

Then we can write, in a logarithmic way,

Q = x i w; where w = ~ a i - - + 2 , aidxi; a i E f f x e

iEA Xi i~A

and g.c.d.(ai; i= 1 . . . n)= 1. The adapted multiplicity I~(~, E; {P}) of ~3at P is defined by /~(~d, E; {P}) = min{v e (ai); i E A} O {ve(ai)+ 1; i ~ A}

where ve (ai) denotes the order of ai at the point P. It generalizes the order of the strict transform of a hypersurface, in the case that we begin with ~ = d f . The main result in [5]

and [7] is stated as follows:

REDUCTION THEOREM ([5], [7]). Let ~ be a non-dicritical holomorphic foliation over X = ( C 3, 0). Then there is a finite sequence o f permissible blowing ups

(2) X = X(1) ~-~)X(2) ~<z) <_-... ~(N~-) X(N)

such that Iz(~(N),E(N); {Q})<.I for each point QEX(N); where ~ ( N ) is the strict transform o f ~ and E ( N ) c X ( N ) is the exceptional divisor o f (2).

Assume now that d i m X = 3 and fix a point P E E . We want to define the statement:

P is a simple singularity o f ~. Put e=e(E,P); we have three possibilities e = l , 2 or 3.

(See Figure 5.)

In the case e = 1, we say that P is a simple singularity iff ~ is an analytic cylinder over a two dimensional simple singularity with e = 1. In particular, in this case Sing ~ is

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, / / / /

Sing ~ Fig. 6

locally a nonsingular curve contained in E and the formal separatrix of the two dimensinal case produces a formal separatrix

Se

at P. (See Figure 6.)

In the case e=2, we have two kinds of singularities. The first kind is locally an analytic cylinder over a two dimensional simple singularity with e=2. In this case Sing ~ is locally the intersection of the two components of E and the only separatrices of ~ at P are the irreducible components of E. (See Figure 7.)

Before defining the simple singularities of the second type with e=2, let us consider the case e=3. Then P is a simple singularity iffkt(~, E; {P})=0 and the singular points near P are simple singularities of the first kind with e=2. The singular locus is the union of the intersections of two components of E and the only separatrices of ~ at P are the irreducible components of E. In order to verify i f P is a simple singularity it is enough to look at any generator of

@=~e~x,~,.

Hence, it is a formal definition. (See Figure 8.)

Now, assume that e=2. Then P is a simple singularity of the second kind with e = 2

E

Sing

Fig. 7

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Sin

E

Sing ~3 f E

Fig. 8

Sing

iff there is a nonsingular formal separatrix Se at P such that E O Se is a (formal) normal crossings divisor at P and P is a simple singularity for ~, relatively to EU St, (i.e. with e=3). (See Figure 9.)

In particular, Sing ~ is the union of the intersections of two components of E USe and St, is the only formal separatrix of ~ at P which is not a component of E. Moreover, in this case, the singular points near P are either simple singularities with e = 1 or simple singularities of the first kind with e=2.

The simple singularities and their normal forms are studied in Part II. First of all we define the pre-simple singularities by the following conditions:

(a) Adapted multiplicity less or equal than one.

(b) The directrix (if it exists), has dimension two and has normal crossings with the divisor E.

The directrix is a geometrical invariant which plays a role similar to the strict tangent space of Hironaka [13] (it is defined in [5], [7] and also in the Part I). Hence,

p / Sing

Sing ~3 f E

Fig. 9

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being a pre-simple singularity is a very geometrical property. Actually, this property is semicontinuous in an evident sense (see Proposition 1.2.6). Let us note however that the semicontinuity depends on the non-dicriticalness property.

Now, let P be a pre-simple singularity. Put

~(~g)p = {DEOx, p;~(D) = 0} = Ox, p

where Ox, e is the (~x,e-module of the formal vector fields. Then we can find two commuting formal vector fields DI,DE in ~((g)r which produces also @p by duality.

Moreover, one of the following situations occurs:

(A) D~=O/Oz, D2=xO/Ox+a(x,y)O/ay; with a(O,O)=O. (In this case DI gives the local analytic triviality.)

(B) Dl=x O/ax+a(x, y, z) a/Oz, D2=y a/Oy+b(x, y, z) O/ay; with a(0, 0, 0)=b(0, 0, 0) =0.

Then, P is a simple singularity iff the eigenvalues of D~ (in case (A)) or of D1, D2 (in case (B)) are non-resonant in a similar sense to the two-dimensional case (quotients not in Q+). These are diophantic conditions, easily reached after finitely many permissible blowing-ups, if we begin with only pre-simple singularities (see Part I I I , w 1).

The fact that D1 and D2 commute allows us to make a simultaneous jordanization of D1 and D2. In particular, we can find a regular system of parameters (x, y,z) of ~x.e in which the semisimple parts of D1 and D2 are diagonal. After a little additional work we can write down formal normal forms for the pre-simple singularities (see Proposition II.4.4). More particularly, in the case of simple singularities we see that a generator f2 of ~e may be written down either in one of the normal forms (i), (ii), (iii) (in the case (A)) or in one of the following normal forms (in the case (B)):

(iv) f~=xyz(a dx/x +fl dy/y+ dz/z); with a .fl*0 and - a , -fl, -a/fl ~ Q+.

(v) Q =xyz.z~(dx/x+fl dy/y +(e+ 1/z s) dz/z); with s>~ 1, O* -/3 ~ Q+.

(vi) g2=xyz(yPzq) s (dx/x +fl dy/y+(e + 1/(y p zq) ~) (p dy/y+q dz/z) ); s ~ > 1, g.c.d.(p, q)= 1.

(vii) Q=xyz "(xPyqzr)S (dx/x + fl dy/y + (e + 1/xPyqzr )~) (p dx/x + q dy/y + r dz/z) ) ; with s ~ > 1, g.c.d. (p, q, r)= 1.

Many of the properties we need from simple singularities can be obtained either directly from the formal normal forms, either from the way we obtain the formal normal forms. For instance, the uniqueness property of the formal separatrix Sp, the shape of the singular locus or even the fact that Sp is "convergent" along the exceptional divisor E.

In Part III, we give a proof of the Desingularization Theorem. By the Reduction Theorem, we may assume that we start with adapted multiplicity less or equal than one. The first thing we do is to prove that we can get only pre-simple singularities after finitely many permissible blowing-ups. This is quite difficult, but most of the technics in

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10

E(N)

F. CANO A N D D. CERVEAU

%

Fig. 10

E(N)

[7] remain valid. Hence we only give in detail those parts which are either different or may be simplified with respect to the general technics in [7]. Once we have only pre- simple singularities, we finish with a computation of "killing resonancies" along the irreducible components of Sing ~.

In Part IV we prove the existence of a convergent separatrix for a non-dicritical holomorphic singular foliation ~ over X=(C3,0). We begin by taking a desingularization sequence like (1). Now, consider the set

0//= U(Y; Y is an irreducible component of Sing ~ ( N ) which is generically contained in only one irreducible component of E(N)}.

Let us fix a connected component q/j of 0~. (See Figure I0.)

Then we have a formal separatrix SQ at each point Q E ~j. Assume for a moment that

SQ

is convergent. By analytic triviality we may continue in an analytic way

SQ

to the points Q' of ~//j with

e(E(N), Q')<~e(E(N), Q).

Hence, the only difficult case is to continue

SQ

to the points Q' with

e(E(N),

Q')=2, but this can be done (see Proposition II.5.5). Thus, we can "glue" the

SQ

in order to obtain a closed hypersurface

Sj(N)cX(N)~which

gives locally a separatrix at each point. (See Figure 11.)

Now because of the properness of the sequence (1) then

S:(N)

projects over a convergent separatrix

SjcX

of the foliation ~. It remains to show that there is at least one q/j supporting a convergent separatrix as above. This is done by taking a non-

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 11

9/j S~(N) Sj(N)

~"'" .."""" 7...

E(N) -"

Fig. 11

E(N)

degenerate plane section of 5~; by [4], the two-dimensional section has at least one convergent separatrix F. Without loss of generality we may assume that the strict transform F(N) of F under (1) is nonsingular and passes through a point Q ~ E(N) with e(E(N), Q)=I. (See Figure 12.)

Now, by analytic triviality, we see that S O is convergent. Thus, the desired 9/j is the connected component of 9/passing through Q.

More precisely. Let us denote by ,Y(N) the formal completion of X ( N ) along the inverse image of the origin zt-l(0), where :t=:r(1)o ... o:r(N). The nature of the formal separatrices S O is of such kind that we can construct a coherent hypersurface

Sj

^{(N) r}

X(N)

F(N)

SQ

Sing (g(N)

(N)

Fig. 12

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12 F. CANO AND D. CERVEAU

supported by each connected component q/j of q/, such that Si(N) gives the separatrix SQ at each point QE r Once again, the properness of ~ assures that Sj(N) projects over a formal separatrix Sj of q3 at the origin. In this way we obtain a bijection

{formal separatrices of @ at the origin ~ {connected components q/j of q/}.

Open questions and related problems.

We give here a list of unsolved problems which seem us to be important ones:

(1) Desingularize holomorphic foliations in higher dimensions and in the dicritical case.

(2) Desingularize a vector field which is tangent either to one or two different foliations of codimension one.

(3) Call "singular holonomy" along q/i the representation z q ( ~ F Sing 0//j, O)___~ Trdiff(~3(N), Q)

where Trdiff(~(N), Q) means the diffeomorphisms of the restriction of ~3(N) to a transversal two-dimensional section at Q into itself. The problem is to understand the non-dicritical singular foliations with the data of the singular holonomy and the holonomy of the components of the exceptional divisor. Some results in this direction may be found in [3] and [19].

(4) Say that q3 has the property 3~ iff it is possible to desingularize q3 by only blowing-up points (and an " a posteriori" eventual addition of irreducible components for

E(N)).

In [11] there is a description of such ~ which are desingularized after one blowing-up. These foliations have first integrals of Liouville type E;ti Log f-. The problem is to describe the foliations having the property ~.

(5) Classify the non-dicritical singular foliations in (C 3, 0) generated by one 1-form with initial part of the type

x dx.

(6) Moduli for simple singularities. As in dimension two, kit is a natural and fascinating problem (see [17]). For example, it is possible to establish some theorems

"Poincarr--Siegel" [21] in the "non-resonant" cases (the eigenvalues ratio are not in Q or even in R) (see [10], [I I]). In the resonant case, there is a rigidity result which is a consequence of Ecalle's Theory (see [12]): assume that ~2 is formally conjugated to

t~=xyz.(xPyqzr)s(dx +fldY (xP;qzr)

)(p.._x_dX

+ q dyy + r dZ))z

then, for generic values of the parameters, ~ is holomorphically conjugated to ~ ([9]).

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 13 (7) Give the topological classification of simple singularities with a given formal normal form.

Acknowledgements. We wish to express our gratitude to J. Martinet and R.

Moussu for very valuable discussions about the subject. One of us (Cano) is very grateful to the Universities of La Bourgogne, Louis Pasteur and Rennes I for his staying there during the preparation of this work. We also thank the referee for his suggestions.

First author partially supported by the DGIGYT.

Second author partially supported by the CNPQ (IMPA).

Part 1. Preliminaries w 1. Adapted singular foliations

Most of the concepts and results in this paragraph may be found in [5], [7].

Let X be a nonsingular connected analytic space over C of dimension n. Fix a normal crossings divisor E of X (always with reduced structure). Let us denote by f~x[-E] the sheaf of germs of meromorphic differential l-forms over X having at most simple poles along E.

DEFINITION 1.1. An adapted to E singular foliation of codimension one over X is a pair (~, E) where ~ is an (Tx-submodule o f f2x[-E] such that:

(a) o ~ is locally free of rank one.

(b) o~^d~=0, where d is the exterior differential.

(c) The quotient f~x[-E]/o ~ is torsion-free.

Let Je be the sheaf of ideals defining E c X . Fix a point P of X. We can choose a regular system of parameters (Xl ... x,,) of the local ring ~?x,e such that

for a certain set A c { 1 ... n). Then, a basis of the stalk f2x, e[-E] is given by

(1.2) ~ dxi~

U {dxi}i~ a.

I. Xi ) iEA

Hence, ~e is generated by a meromorphic differential 1-form

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14 F. CANO A N D D. CERVEAU

dx i

( 1 . 3 ) to-- Z a i - - + 2~aidxi; a i r , x , P.

lEA Xi i~A such that to A dto =0 and g.c.d.(ai; i= I ... n) = 1.

In the case E=~3, we find the usual notion of singular foliation of codimension one (cf. [I1]). Let us denote by ~ ( X , E ) the set of adapted to E singular foliations of codimension one over X. Then, we have a bijection

(1.4) hol: ~(X, E)---~ ~(X, ~)

which is defined by the following property:

(1.5) If ((~, ~3) = hol((~, E)), then ~lx-e -- ~lx-e.

Moreover, if ~e is generated by 09 as in (1.3), then ~e is generated by

o r

where the set A* is given by

(1.7) A* = (iEA;xi does not divide ai}.

Now, fix (c~,~3)E~(X,~) and a point P E X . Assume that ~e is generated by f2 E f2x, v. We recall that a "separatrix", respectively a "formal separatrix", of (~, ~) at P is a principal prime ideal f(~x,v, respectively f&x,e, such that

(1.8) f d i v i d e s f2^df.

(cf. [I1]). Here ~x,e denotes the completion of ~x,e along the maximal ideal. An

"invariant analytic space" of (~, ~3) is an irreducible closed analytic space K of X such that

(1.9) f2lr = 0

at the nonsingular points P of K. Any invariant analytic hypersurface H c X of (@, ~) defines a separatrix at each point P E H. Conversely, an irreducible hypersurface H c X defines an invariant analytic space of (~, ~) iff it defines a separatrix at a point P E H.

Let ( ~ , E ) E ~(X, E) and fix an irreducible component F of E. We say that F is a

"non-dicritical component" of E for ( ~ , E ) iff F is an invariant analytic space of hol((~, E)). Otherwise, we say that F is a "dicritical component" of E for (~:, E). Then,

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taking the notation of (1.7), we have that

(1.10) A* = {iEA; (xi=0) is a non-dicritical component for (if, E)}.

DEFINITION 1.2. Given ( ~ , E ) E ~ ( X , E ) and a point P 6 X , the adapted order v ( ~ , E ; P ) is the M-adic order o f the submodule ~p o f •x,e[-E], where Jig is the maximal ideal o f (Tx,~,. The singular locus Sing(if, E) is the set o f the points P 6 X such that v( ~, E;P)~>I.

With the notations of (1.3), we have that

(1.11) v(~, E, P) = min{ve(ai); i = 1 ... n).

Where ve(ai) is the M-adic order of ale ~Tx, e. The singular locus is a closed analytic subset of X and since Q x [ - E ] / ~ has no torsion, we have

(1.12)

If hol((~, E))=(c~, 6), note that (1.13)

Codimx Sing(o ~, E)/> 2.

Sing(~, E) c Sing(Y, 0) and we also have that CodimxSing(~, O)/>2.

Let Y c X be a nonsingular analytic subspace of X having normal crossings with E.

Let

(1.14) sr: X'---~ X

be the blowing-up with center Y. Put E'=er-t(EU Y), with reduced structure. Then E ' c X ' is also a normal crossings divisor of X'. Now, fix ( ~ , E ) f i q ~ ( X , E ) and put (~, O)=hol((~, E)). Then there is a unique ( ~ ' , E') in ~(X', E'), respectively ((g', 6 ) in

~(X', O), such that

(1.15)

~'lx,-=-,r

= "~lx-r, respectively qd'tx,_,~-,(r ) = ~glx-r, under the isomorphism x: X ' - z - * ( Y)--->X - Y. Moreover

(1.16) ( ~ ' , 6) = hol((o~', E')).

(cf. [5], [7]).

D E F I N I T I O N 1.3. In the above situation we say that ( ~ ' , E ' ) is the adapted strict transform o f (o~,E) by x and that ((g', 6 ) is the strict transform o f (~,O) by x.

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Let Jrc~? x be the sheaf of ideals defining Y c X . Denote by Ox[E* Y] the sheaf of germs of vector fields being both tangent to E and Y. Let 0//(~:, E; Y) be the image of the bilinear mapping

(1.17) ~ • Ox[E* r] ~ ~?x

given by (to, D)~to(D).

DEFINITION 1.4. The adapted multiplicity/~(~,E; Y) o f ( ~ , E ) at Y is the Jy-adic order o f ~ E; Y).

Remarks 1.5. (a) Take P E Y and P' E er- t(y). Let to be a generator of ,~e and l e t f b e a reduced equation of the exceptional divisor :r-l(y) at P'. Put a =/~(~, E; Y). Then J;'p, is generated by f-azt*to (cf. [5], [7]).

(b) Since Y has normal crossings with E, we can find a regular system of parameters (Xl ... x~) of ~?x, P and two sets A, B c (1 ... n} such that

If co generates ~? as in (1.3), we have explicitly that

(1.19) p ( ~ , E; Y) = min({vy(ai); i ~. B - A } U {vy(ai)+ 1; i E B - A } )

where vr(ai) denotes the J~,,e-adic order of aiE ~x,e. In particular, we can compute

~(~, E; Y) at any point P E Y.

(c) The adapted multiplicitity generalizes in a natural way the usual multiplicity of a hypersurface and its behaviour under blowing-up (cf. [7], Introduction).

Consider a point P E Y. Denote by Q(o~,E; Y, P) the 2d-adic order of the ideal

~(g*,E; g)ecOx, e, where ~ is the maximal ideal of ~x,e. More explicitly 0 ( ~ , E ; Y;P)=min({ve(ai);i~i B - A } U {ve(ai)+ l ; i E B - A } ) . (1.20)

Note that

(1.21) Q(~:, E; Y;P) ~>~t(~,E; Y)

and the equality holds outside an analytic subset W of Y, with W~= Y.

DEHNIXION 1.6. Let Y be an irreducible closed analytic subspace o f X and let (~, E) E ~(X, E). Fix a point P E Y. Then Y is a permissible center for (~, E) at P iff the following properties hold:

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 17 (a) Y=Sing(~3, ~), where (~, ~)=hol((~, E)).

(b) Y is nonsingular at P and has normal crossings with E at P.

(c) The equality Q(~, E; Y, P)=/~(~, E; Y) holds.

A permissible center Y for (~, E) is a permissible center at each point P ~ Y.

Remarks 1.7. (a) A point {P}cSing(~, ~) is a permissible center.

(b) An analytic subspace YcSing(~, ~3) is a permissible center outside an analytic subset W of Y, with W=~ Y.

Now, we are able to define the "non-dicritical singular foliations". Here we shall give a technical definition which is convenient for our purposes. Another characteriza- tions of this condition are given in [6].

DEFINITION 1.8. We say that ( ~, E) E ~(X, E) is non-dicritical iff there is no fin#e sequence

(1.22) {X(/), E(O, ~(i), ql (i), Y(i), at(i+ I)}i=0,1 ... U such that:

(a) X(0)=X, (~(0), E(0))=(ff, E).

(b) For each i=0, 1 ... N we haoe that:

(bl) ~ is a nonempty open set.

(b2) Y(i)c~ is a permissible center for (~(i)l~r ql(i)), (b3) zc(i+ 1):X(i+ l)--*ql(t) is the blowing-up with center Y(O.

(b4) (~(i+ l),E(i+ l)) is the adapted strict transform o f (~(i)l~ o, E(t~ N ~(i)) by at(i+ 1).

(c) There is a dicritical component o f E(N) for ( ~ ( N ) , E ( N ) ) .

In particular, making N = 0 , we see that if (,~, E) is non-dicritical, then E has no dicritical components for ( ~ , E ) . That is, each irreducible component of E is an invariant hypersurface for hol((~, E)).

TrIEOREM 1.9 (Stability Theorem). Let (~, E) E ~ ( X , E) and let Y c X be a permis- sible center for (~, E). Let at: X'---~X be the blowing-up with center Y and let ( ~ ' , E') be the adapted strict transform o f (if;, E) by at. Fix a point P E Y and a point P' E at- I(P).

Then:

(a) v(J;', E'; P')<~v(~, E; e).

(b) I f (~, E) is non-dicritical, then I~(~', E'; {P'})~</t(~, E; {P}).

Proof. [7], Theorem 1.2.7; Theorem 1.3.3. []

2-928285 Acta Mathematica 169. Irnprim~ le 20 aoflt 1992

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Hence, in the non-dicritical case, we can use the invariant (1.23) (r, m) = (v(~, E;P),I~(~T, E; {P}))

in order to control the behaviour of the singularities under permissible blowing-ups.

Let Y c X be a permissible center for (~, E) E ~(X, E). Fix a point P E Y. Then (1.24) v(~, E; P) ~< ~(~, E; Y, P) =/~(~, E; (P}) <~ v(~, E; P)+ 1.

We say that Y is "appropriate" at P iff

(1.25) /~(~, E; Y) =/~(,~, E; {e}).

PROPOSITION I. 10 (Stationary sequences). Let ( ~, E) E ~ (X, E) be non-dicritical.

Put ( ~ , ~ ) = h o l ( ( ~ , E ) ) . Fix an irreducible curve FcSing((g,Q) and a point P E F . Consider an infinite sequence

(1.26) {X(i), E(i), ~(i), F(t~, P(i), :t(i+ 1), r(t3, m(i)}i>~o defined as follows:

(a) X(0) =X, (~(0), E(0)) = (~, E), F(0)=F, P(0)=P.

(b) ~r(i+ 1): X(i+ 1)--->X(i) is the blowing-up with center P(i).

(c) F(i+ 1) is the strict transform o f F(t) by ~r(i+ 1).

(d) P(i+ 1) E F(i+ 1) N ~t(i+ 1)-~(P(i)).

(e) ( ~ ( i + 1), E(i+ 1)) is the adapted strict transform o f (~(i), E(13) by Jr(i+ 1).

(f) r(/)=v(~(0, E(i);P(i)), m(t3 =/t(~(t3, E(i); {P(i)}).

Then, the following two conditions are equivalent for any index N:

(A) F(N) is nonsingular and has normal crossings with E(N) at P(N) and for each i>~N we have that (r(i), m(i))=(r(N), m(N)).

(B) F(N) is permissible and appropriate for ( ~ ( N ) , E(N)) at P(N).

Proof. [7], Theorem II. 1.1. []

Remark 1.11. There is always an index N~>0 such that the above condition A is satisfied. Hence, if F is not permissible at P, we can achieve this condition by blowing- up the point P finitely many times.

Now, we can state the main result in [5] and [7] as follows:

THEOREM 1.12 (Reduction Theorem). Assume that dimX=3. Fix a non-dicritical (~, ~3) E ~(X, 0) and a point P E Sing(~, ~). Then there is an open set X(O) o f X,

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SEPARATRICES A N D D E S I N G U L A R I Z A T I O N OF FOLIATIONS 19 P EX(O), and a sequence o f permissible blowing-ups

~r(1) ~(2)

(1.27) X(0) ~ X(1) *-- ... ~lV)x(N),--- such that

(1.28) Iz(,~(N),E(N);(Q))<~ 1, for all QEX(N),

where ( J;(N), E(N)) is the adapted strict transform o f ( ~lx~o ), 6) under the composition

~r(1)o... oar(N). Moreover, the sequence (1.27) may be taken in such a way that

(1.29) {P} = center ofar(1),

(1.30) Singhol((~(N), E(N))) = E(N).

Remarks 1.13, (a) Put ar=~r(1)o...oar(N). Then (1.29) means that Z(N)=ar-1(P) is also a normal crossings divisor Z ( N ) c E ( N ) .

(b) In [5] and [7], the above theorem is stated in terms of the germs of X at P. In particular, the condition (1.28) is stated only for the points Q E ar-1(P)=Z(N). Neverthe- less, the semicontinuity of the adapted multiplicity in the non-dicritical case ([7], Remark 1.1.6) allows us to state the result in terms o f a n open set X(O)cX, PEX(O).

w 2. Pre-simple singularities

Before defining pre-simple singularities, let us recall the notion of "directrix" intro- duced in [5], [7]. Actually, we shall only consider here the case of adapted order equal to one, which is simpler than the general case.

Given an element f E t~x,e and an integer s~>0 such that v e ( f ) ~ s , let us denote by InS(f) the image of f in ~s/~s+l, where ~ is the maximal ideal of ~Tx, e. Actually InSfE Gr(~Tx, e), where Gr(t~x,e) is the graded ring for the ~t-adic filtration of tTx, e. Note that Gr(e~x,e) is a polynomial ring in the indeterminates Xi=In~(xi), i= 1 ... n.

Now, consider a non-dicritical (4, E) E ~(X, E) a point P E X such that

(2.1) m =/z(~, E; {P}) ~< 1.

Note that if r--v(~,E;P), then r<<-m<-r+l and hence either r=0 or r = l . Assume that r= 1 and let us write a generator to of ~e as in (1.3):

dxi Z ai dxi; a i E ~x P.

(2.2) to = Z a i - - +

iEA X'i i~A

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20 F. C A N O A N D D . C E R V E A U

Then, the directrix Dir(~, E; P) o f (g~, E) at P is defined by (2.3) Dir(~, E;P) = t'l (Inl(ai) = 0) ~- TeX,

i E A

where TeX is the tangent space o f X at P. (In the case r = 0 , the directrix is not defined.) Denote by JDir(~, E; P) the ideal defining the directrix. Then

(2.4) JDir(ff, E; P) = E Inl(ai ) Gr(Ox, e),

i E A

P R O P O S I T I O N 2.1. In the above situation, assume that r = m = l . Let Y c X be a permissible center for (~, E) with P E Y. Let zl: X'--->X be the blowing-up with center Y

and let (3, E) be the adapted strict transform o f (~, E) by :t. Then:

(a) TeYcDir(,~, E; e).

(b) Let P' E:t-1(P) be such that v(~',E';P')=Iz(J;',E'; { P ' } ) = I . Then:

(2.5) P ' E Proj(Dir(~, E; P)/T e Y)c-Proj(TeX/T p Y) = ~-l(p).

Proof. [7], Theorem 1.4.8 (see also [5], Theorem 4). []

L e t us denote by e=e(E, P) the number of irreducible components of E through P.

(Actually e(E, P) is the multiplicity of E at P, moreover, with the notation of (1.1) we have e=#A.) Take a codimension one vector subspace H of TeX. We say that H has normal crossings with E iff there are e + 1 independent linear forms tp0, tpl ... ~0 e on TeX such that

(2.6) H = (q00 = 0).

(2.7) TpEi=(q~i=O), i=l ... e,

where E l . . . Ee are the irreducible components of E at P.

DEFINITION 2.2. Consider a non-dicritical (3, E) E ~(X, E). Put ( ~, ~)=hol((3;, E)) and consider a point P E Sing(~, 9). We say that P is a pre-simple singularity for (3, E) iff one o f the following conditions holds:

(a)

v(o ~, E;P)=O.

(b)/z(~, E; (P})=v(~;, E; P)= 1 and Dir(~, E; P) has normal crossings with E. (In particular, the dimension o f Dir( ~, E;P) is n - 1 . )

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Remarks 2.3. (a) If P is a pre-simple singularity, then necessarily e(E, P)~>I. In fact, if e(E, P) =0 and v(o~, E; P) =0, then P ~ Sing(Y, 9); if e(E, P ) = 0 and v(~, E; P) = I, then/~(~, E, {P})=2.

(b) I f P is a pre-simple singularity with e(E, P)=n, then necessarily v(~, E;P)=0. In fact, if the directrix exits, it cannot have normal crossings with E.

LEMMA 2.4. Consider a non-dicritical (~, E) E ~(X, E). Let P E X be a point such that

(2.8) v(o ~, E; P) =/t(~, E; {P}) = I.

Let F be the intersection o f all the irreducible components o f E through P. Then P is a pre-simple singularity f o r (~%, E) iff

(2.9) Dir(o%, E;P) :~ TeE.

Proof. The "only i f " part is trivial. Conversely, assume that (2.9) holds. Choose a regular system of parameters (xl ... x,) of 6x.v such that

Then a generator co of ~e is written down as follows:

(2.11) c o =

E

a, dx' + E a , d x , 9

i= 1 . . . e X i i > e

Denote by Ai=Inl(ai), i = 1 . . . n; X i = I n l ( x i ) , i = 1 . . . n. We can assume without loss of generality that A 1 =Xe+ v Now, the integrability condition co ^dco=O implies that

0.4 ~ 0.41

(2.12) A1 axe+ 1 - As OXe+---1 - As' s = 2 ... e.

Hence A~=2sX~+I, 2~E C, for all s=2 ... e and thus

(2.13) Dir(~;, E; P ) = (X~+~ = 0).

This ends the proof. []

PROPOSITION 2.5. Consider a non-dicritical ( J ; , E ) E ~ ( X , E ) . Let P E X be a pre-simple singularity f o r (~, E). Consider a permissible center Y c X f o r (~, E).

Let n:X'--~X be the blowing-up with center Y and let ( ~ ' , E ' ) be the adapted

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strict transform o f ( ~ , E ) by yr. Put ( ~ ' , O ) = h o l ( ( ~ ' , E ' ) . Then each point in

~r-l(P) N Sing(~d', ~) is a pre-simple singularity for ( ~ ' , E ' ) . Proof. By Theorem 1.9, the only bad case is P' Ezr-~(P) with (2.14) /x(~', E'; ( P ' ) ) = v ( ~ ' , E ' ; P ' ) = 1.

Let F be the intersection of all the irreducible components of E through P as in Lemma 2.4. Define

(2.15) d(~, E; P) = dimc((JDir(~, E; P)+JTvF)/JTvF) ~

where J T v F is the ideal of TvF and the subindex 1 means "linear part" (of. [7],I,(4.2.4) or [5],w 4). Since P is a pre-simple singularity then

(2.16) d(~, E;P) = 1.

Now by [7], Theorem 1.4.8 (c), or [5], Theorem 4(iv), we have that (2.17) d ( J ; ' , E ' ; P ' ) >! d(,~', E;P) -- 1.

This implies that the condition (2.9) of Lemma 2.4 holds. Hence P' is a pre-simple

singularity. []

PROPOSITION 2.6. Consider a non-dicritical ( ~ , E ) E ~ ( X , E ) . Put (~,~)=hol((~,E)).

Then the set

(2.18) Sing*(o ~, E) = ( P E Sing(Y, ~); P is not a pre-simple singularity) is a closed analytic subset o f X.

Proof. It is a local statement. Fix P E X and let (X 1 . . . Xn) be a regular system of parameters of Ox, e such that

(2.19) E = ( i - ~ x i = O ) , locally at P.

\iEA /

Let us consider a generator w of ~e given as in (1.3) by

dxi

E a i d x i . ~ a i ~ x p .

(2.20) w = E a i - - +

iEA Xi il$A

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S E P A R A T R I C E S A N D D E S I N G U L A R I Z A T I O N O F F O L I A T I O N S

Now, let us put (2.21)

Given A' c A , let us define the closed analytic sets (2.22)

(2.23) (2.24)

F ( s ) = { Q ; e ( E , Q ) ~ s } , O<.s<.e(E,P).

E A, = A ( x i = O) iEA'

C A, = S i n g ( ~ , E)fl {Q;(Oai/Oxj)(Q ) = 0, i E A ' , j q . A ' } D A, -- a d h e r e n c e o f C A, fl (E a , - F(1 + #A')).

In view o f L e m m a 2.4, we have that

(2.25) Sing* (~, E ) fl (E A,-F(1 + ~*A')) = C A, O (E A,-F(1 + #A')).

Now, it is enough to prove that

(2.26) I f A ' c A " , # A " = # A ' + I , then DA, N(EA,-F(I+#A")) cDa,,.

Since in this case (2.27)

23

Sing*(~, E ) = t9

DA,.

A ' c A

In order to prove (2.26), let us r e a s o n b y contradiction, assuming that (2.26) is not true.

We can assume without loss o f generality that (2.28) A" = a = ( 1 , 2 . . . e}.

(2.29) A' = {2 . . . e}.

(2.30) T h e r e is a point P E D A,-D A.

Then, we can find an analytic b r a n c h F at P such that F c D A, and FCEA. B y Proposition I. 10 and Proposition 2.5, blowing-up the point P r e p e a t e d l y , we may assume without loss o f generality that F is a permissible c e n t e r for (~:, E). H e n c e we can take coordinates such that

(2.31) F = (x2 = . . . = x , = 0).

The fact F c S i n g ( ~ , E ) implies that vr(ai)~l, i = I . . . n. Moreover, since P is a pre-

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simple singularity, then

(2.32) 1 ~<#(~, E; F) ~<#(,,~, E; {P}) = I.

H e n c e / , ( ~ , E; F)-- 1. A s s u m e first that

(2.33) Vr(ai) I> 2, for i = 2 . . . e.

Then, in view o f R e m a r k 1.5 (a), we find a dicritical c o m p o n e n t by blowing-up the center F. Contradiction. Thus, we may assume that

(2.34) Vr(ai; i = 2 . . . e) = 1.

Since F C D A , , we can write

(2.35) a i = ~ cpo.(xl)xi+u)i, i = 2 . . . . , e, j=2,...,e

where Wi E (x2 . . . x,) 2. Moreover, since P is a pre-simple singularity we can a s s u m e without loss o f generality that

(2.36) Inl(al) = Xe+ I = Inl(Xe+l).

NOW, looking at the coefficient o f d x 1 ^ d x s ^ d x e + l, s = 2 . . . e in the integrability condition co^de0=0, we have that

[ 8a, 8ae+ ,\ / 8ae+ , 8a, "~ ( 8al a aa, "~

(2.37) X l / - - - a e + l + a s - - I + x s t a l

\ OX 1 aX 1 ] \ aX s ~ x s a e + l ) + \ a X e + l a s - 1 aXe+l: = 0 . Looking at the terms o f o r d e r o n e with r e s p e c t to (x2, . . . , x , ) in (2.37), we find that (2.38) ~ q~sj(xOx:=O, s = 2 . . . e.

j=2, ...,e

Then (2.33) holds and we find a contradiction as above. []

Part II. Simple singularities and their normal forms w 1. Formal normal forms for abelian Lie algebras of vector fields

H e r e we shall recall some e l e m e n t a r y facts a b o u t the t h e o r y o f formal normal forms for vector fields and abelian Lie algebras o f v e c t o r fields. Since these results are well

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 25 known, we shall only sketch the proofs. A good reference about this subject is Martinet's Bourbaki [16].

Let X be a nonsingular analytic space over C of dimension n. Denote by Ox the tangent sheaf of X. Given a point P EX, we shall denote by Ox, e the ~-adic completion of Ox, e, where ~ is the maximal ideal of 6x, e. The elements of Ox, e are called formal vector fields at P. They induce derivations of ~x,e in an obvious way. Moreover, we have a canonical inclusion

(1.1) Ox, e C (gx, p.

Put ~ = ~ x , e . Given a formal vector field D E ~Ox, e and an integer k~> 1, we have an induced derivation

(1.2)

Dk: j///j//k+l ___~ ~///~//k+ 1 f + ~l k+l ~ D ( f ) + jtt k+l.

(Note that A/(A)k+l=~/~/k+l.) We can put the C-linear operator D k in its Jordan normal form. That is, there is a unique pair of linear operators Dks and DkN, being respectively semisimple and nilpotent, such that

(1.3) D k = Dks+DkN; DksDkN--DkNDk s = 0

Actually both Dks and D ~ are derivations of ~ / ~ k + l as ~x,e-module (to see this, it is enough to compute Dks and DkN directly in terms of coordinates). By uniqueness of the Jordan decomposition we can take limits

(1.4) D s = limkDks; D N = limkDkN

which are formal vector fields at P such that

(1.5) D = Ds+DN; [D s, DN] = 0

where [.,.] denotes the Lie bracket.

The decomposition of (I .5) is called the Jordan decomposition of D. We say that D is semisimple, respectively nilpotent, if D=Ds, respectively D=DN.

PROI'OSIXION 1.1. Consider a semisimple formal vector field DE~lOx, p. Let (x'i, ...,x's) be a ~l-regular sequence in ~x,e such that

(1.6) D(x'i)=2'ix'i; 2'iEC, i = 1 ... s

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for a certain s, with O<-s<,n. Then there is a regular system o f parameters (xl . . . x~) o f (Yx, e such that

D(xi)=~,ixi;

/],i E C, i = 1 . . . n (1.7)

'" ~,i=J.i, t i = l , s.

X i ~ X i~ " " '

Proof. The x'i, i-- 1 ... s, gives a part o f a basis of eigenvectors for D k. N o w , it is enough to complete it and to take limits w h e n k ~ o o . []

In the situation o f the above Proposition 1. I, any regular system o f p a r a m e t e r s (Xl ... x,) o f ~x, e satisfying (1.7) is said to be a linearizing formal system for D.

Remark 1.2. Given D E d ~ g x , e, then D is nilpotent iff D l is nilpotent. This is evident, since the hi, i= 1 . . . n, o f (1.7) are the eigenvalues o f Dis.

Consider D E ~ ( g x , e. L e t (xl . . . x,) be a linearizing formal s y s t e m for Ds and let 3,=(21 ... 2,) be the c o r r e s p o n d i n g eigenvalues. Take the following notations

(1.9) If I = (i I . . . i.) E N", then x I = x I ... ^{i I}

x . .

^{i n}

(1.10) ( L I ) = ~

~jij,

[II= ~ ij.

j = l , . , . n j = l . . . n

Now, in view o f (1.7), we have that

(1.11) [Os, x l a-'-~-] = ((,3.,I)--,~i)xlo--~l [ OxiJ ; i = 1 . . . n.

That is, the monomial formal v e c t o r fields x1(O/axi), i= 1 ... n, are eigenvectors for the operator [Ds," ] with eigenvalues ( 2 , 1 ) - 2 i . Now, write

(1.12) ON= E Z at j x I O .

j=1 ... ~l>~l ' Oxj By (1.11), the condition

[Ds,

DN]=0 is equivalent to say that

(1.13) If (2, I) -2j ~: 0, then al, j = O.

H e n c e D = D s + D N can be written down as follows:

(1.14) D = E 2jxj a-~-+ ~ >_ E at, j x ' 0 .

j = l . . . n ~ X j n = l [/1~1; (,t,/)=,!j

OXj

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 27 Remark 1.3. The formula (1.14) has the additional property that the linear operator

(1.15)

DN= ~ X

al,j Xl o

j=l ~/I~>I;Q.,I)=2j ~Xj is actually a nilpotent operator.

The above formula (1.14) may be generalized to finite dimensional abelian Lie algebras of formal vector fields as follows:

PROPOSITION 1.4. Let (~cft(~x,e be a finite dimensional abelian Lie algebra o f formal vector fields vanishing at P. Then:

(a) There exist two finite dimensional abelian Lie algebras (~s and ~ N in ft(gx, e such that:

(al) ~ C ~ S ~ N . (a2) [(~s, (~N] =0.

(a3) If D ~ ~s, resp. D E ~N, then D is semisimple, resp. nilpotent.

(b) Let (X'l ... x's) be a J~-regular sequence in Ox, P such that:

(1.16) For all D E ~ , Ds(x'i)=2'i(D)x'i; ;t'i(D)EC, i= 1 ... s.

for a certain s, O<.s<.n. Then, there is a regular system o f parameters (xl ... xn) o f ~Yx, p such that:

(1.17) For a l I D E @ , Ds(xi)=,~i(D)xi; 2i(D)EC, i = I ... n.

(1.18) x i--

Xri,

2i(D)=,Ui(D); for all D E ~ and i = 1 .... , s.

Proof. Take

(1.19) ~ s = { D s ; D E @ } ; @ N = { D N ; D E @ }.

Thus, we have obviously (al) and (a3). Given Z E ft(gx, e, consider the linear operator [Z,. ], acting on ~ ) x . P - Working as above, we have a unique decomposition

(1.20) [Z,-] -- [Z,- ]s+ [Z,. ]N-

where [Z,. ]s and [Z,-]g are commuting linear operators that produce the semisimple- nilpotent decomposition of [Z,-], modulo (~)k+l, for all k ~ > 1. Moreover, we have that (1.21) [Z,.]s = [Zs,'] and [Z,']N = [ZN,'].

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2 8 F. CANO AND D. CERVEAU

Take two elements D, D ' E @. T h e fact that

(1.22) [D, D'] -- 0

means that

D'

is an eigenvector for [D,. ] with zero eigenvalue. H e n c e it is so for its semisimple part [ D , - ] s = [ D s , . ]. Thus

(1.23)

[Ds,

D'] = 0 = - [ D ' ,

Ds].

Now,

Ds

is an eigenvector for [D', .], h e n c e for

[D's,'],

and

(1.24)

[D's, Ds] -- O.

This proves that (~s is an abelian Lie algebra. F r o m (1.22), (1.23) and (1.24), we d e d u c e that

(1.25)

[Ds,

D'N] = 0; [DN, D'N] = 0.

H e n c e , ~ v is an abelian Lie algebra and (a2) holds.

(b) C h o o s e a regular s y s t e m o f p a r a m e t e r s (X"l .. . . . x",) o f

(~x,e

such that (1.26) Inl(x"i) = Inl(x'i), i = I . . . s.

(1.27)

Dls(Inl(x"i)) =J,i(D)Inl(x"i);

for all D E ( ~ , i = 1 . . . n.

Note that (1.27) is always possible by simultaneous reduction to the J o r d a n form o f a set o f commuting e n d o m o r p h i s m s o f a finite dimensional vector space. In particular, (1.27) allows us to define

(1.28) ~,(D) = (21(D) . . . 2,(D))

for all D E @. Actually 2: D ~ A ( D ) is a linear mapping. L e t us fix an element Z o f ~ , satisfying the following generic property:

(1.29)

(2(Z),I)

= 2 j ( Z ) =~ ( ( 2 ( D ) , I ) = 2 j ( D ) ; for all B E G ) ; for all j = 1 . . . n.

We can take a regular system o f p a r a m e t e r s (xl . . . x,) o f

~x,e

such that (1.30) Inl(xi) = Inl(x"i), i = I . . . n.

n

(1.31)

Zs

= E •i(Z)xi 0

i = l aXi"

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Given D E (~, let us write (1.32)

29

n

D 1 = Z,~,i(D)xi-~-; D 2 = D - D 1.

i~l t d A i

Hence Dl is semisimple, D2 is nilpotent (since D21 is nilpotent) and D=DI+D 2. If we show that [D1, D2] =0 we are done, since then D=D1 +D2 is the Jordan decomposition of D. Note that

(1.34) 0 = [Z, D 1 + D 2 ] --- [Z, D1] + [Z, D2] = [Z, D2].

But in view of the property (1.29) (see also (1.13)), we have that (1.35) [Z, D2]-.O => [ D 1 , D 2 ] = 0 .

This ends the proof. []

In particular, the above (xl ... x,) is a common linearizing formal system for Ds, for each D E @ . Given D E ~ , denote by 2(D)=(21(D) ... 2n(D)) the corresponding eigenvalues, like in (1.28). Take a generic Z E ffl like in the proof above. The condition [Zs, DN]=0 means that D=Ds+DN can be written down as

n ~

(1.36) D = E 2 j ( D ) x j ~ j + E a l j ( D ) x ' a

j = l " j=l l e ~j((~) aXj

where the set ~ ( ~ ) is given by

(1.37) ~,((~) = {IENn;[I[ ~>2, (2(D), I) = 2j(D), for all D E g6}.

Note also that the second term on the right hand-side of (1.36) defines a nilpotent operator.

The formula (1.36) will be a key tool in our study of the normal forms for the pre- simple and simple singularities.

w 2. Dimension two revisited

Let E c X be a normal crossings divisor of X. Denote by Ox[E] the sheaf of germs of vector fields which are tangent to each irreducible component of E. Thus Ox[E]COx and moreover we have a perfect pairing

(2. I) ~ x [ - E ] x Ox[E ] ---, ~x"

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3 0 F. CANO AND D. CERVEAU

Given a coherent submodule ~ o f ~ x [ - E ] , denote by ~ ( ~ , E) the submodule of Ox[E]

which annihilates ~, i.e., the orthogonal of o% under (2.1). Conversely, given a coherent submodule ~ of Ox[E], let ~ ( ~ , E ) c g 2 x [ - E ] be the orthogonal of ~. Consider (o%, E) E ~(X, E), the fact that ~ i s inversible and that ~ x [ - E ] / ~ h a s no torsion implies that

(2.2) ~ - - ~(~(o%, E), E).

Moreover, if (o%, E) E ~(X, E) is non-dicritical and (~, O)=hol((o~, E)), then

(2.3) ~ (~, E) = @(~d, ~) ~- Ox[E].

Given a point P E X, then (2.1) induces a perfect pairing between 5t-adic completions (2.4) ~x,e[ - E ] • Ox, e[ E ] "-~ (~x.p"

If ~ ( . , E) and ~ ( . , E) denote the corresponding orthogonality operators, then (2.5) ~(~z,, E) -- ( ~ ( ~ , E)~) ^ , ~ (~j,, E) = ( ~ ( ~ , E)~,) ^,

(with evident notations), for each coherent ff;Cf]x[-E] and ~COx[X].

Assume now that n = d i m X = 2 and take a non-dicritical ( ~ , E ) E ~ ( X , E ) . Put (~, ~)=hol((~, E)). Consider a point P ~ Sing(Y, ~) which is a pre-simple singularity for (~, E). Then ~ (~, E)e is generated by a single germ of vector field D E Ox, e[E] with

(2.6) D E ( Ox, e[E ]) N d,t. Ox, e.

Moreover, the fact that P is a pre-simple singularity implies that

(2.7) Dis 9 O,

i.e., O 1 has at least one nonzero eigenvalue. Consider an irreducible component F of E at P (it exists since e(E, P)~> 1 by Remark 1.2.3). Let x E ~Tx, l, be a generator of the ideal Jf, e. Since D is tangent to F, then In~(x) is an eigenvector of D 1. Hence

(2.8) Dl(Inl(x)) = Mnl(x), ~. E C.

Let # fi C be the eigenvalue of D ~ corresponding to an eigenvector of D~s independent of In~(x). Note that 0,,~):~(0, 0). Define the invariant A(~, E;F;P) by

(2.9) A(~, E;F;P) = ~//z E C U {~}.

It is intrinsically defined.

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 31 DEFINITION 2.1. In the above situation, we say that P is a simple singularity for (o%, E) iff

(2.10) A(~, E; F;P) ~ Q+

where Q+ = (strictly positive rational numbers).

Remarks 2.2. (a) The above definition does not depend of the chosen irreducible component F of E at P. In fact, i f F b F2 are the two irreducible components o f F at P (in the case e(E, P)--2), then

(2.11) A(o~,E;FE;P) = 1/A(,~,E;F1;P).

(b) The simple singularities are stable under blowing-ups. More precisely, let P be a simple singularity for (~, E), let ~ r : X ' ~ X be the blowing-up with center P and let ( ~ ' , E') be the adapted strict transform of (~, E) by ~r. Put (~d', ~ ) = h o l ( ( ~ ' , E')). Then there are exactly two singular points P'1,P'2 in Sing(~d',~) with P'iEJr-l(P), i=1,2.

Both P'I and P'2 are simple singularities for ( ~ ' , E ' ) . The strict transform of each irreducible component of E at P passes through one of these points. Moreover, fix an irreducible component F of E at P and assume that P'I is in the strict transform F' of F by ~r. Then

(2.12) A ( ~ ' , E ' ; F ' ; P ' ~ ) .--- A ( o ~ , E ; F ; P ) - I (2.13) A ( ~ ' , E'; z r - l ( e ) ; p , i) = 1/[A(~, E; F; P) - I I.

(2.14) A ( ~ ' , E';x-~(e); P'2) = 1/[[ 1/A(~, E; F; P ) ] - 1].

(c) Let P be a simple singularity with e(E, P)=2. Then the only invariant analytic spaces of (~, O) through P are the two irreducible components of E at P.

(d) Let P be a simple singularity with e(E,P)=I. Then (~,O) has exactly two formal separatrices at P. One of them is given by the ideal Je, e of the divisor E, it is of course a convergent one. The other one, say f . dx, v, is non singular and transversal to E (i.e., f jointly with a local equation of E define a regular system of parameters of

~x,e). Classical results say that we can t a k e f t o be convergent in the case that

(2.15) A(o%, E; E; P) * ~ .

(e) Finally, let us recall that Seidenberg's result of desingularization [20] means that in the two-dimensional case we can get a situation with only simple singularities after finitely many blowing-ups.

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32 F. CANO A N D D. CERVEAU

LEMMA 2.4. Let P be a pre-simple singularity for (~, E) as above. Then

(2.16) A(,~, E; F; P) ~ Q + - N U (I/N).

Otherwise (~, E) would be a dicritical singular foliation.

Proof. Let us reason by contradiction, assuming that (2.17) A(~, E; F; P) = p/q E Q + - N LJ (l/N),

where p, q E N, p, q ~ 2 and g.c.d.(p, q)= 1. Now, let us make induction on p+q. We can take a regular system of parameters (x, y) of 6x, e such that a generator ~ of ~de is given by

(2.18) ff~ = (py + cp(x, y) ) dx+(-qx+qJ(xy)) dy

with ve(q~, ~0)~>2. Assume that q<p, let us blow-up the point P and look at the point P' corresponding to the strict transform of (y=0). Putting x=x', y=x'y', a generator t)' of

~3'e is given by

(2.19) Q' = ((p - q ) y' +x'tp'(x', y')) dx' + ( - q x ' +x'2~0'(x ', y')) dy'.

We distinguish two cases:

Case 1: q = m ( p - q ) , for some integer m~>2. In this case, blowing-up P' and looking at the point P' corresponding to the strict transform of x' =0, a generator f~" of ~d"e, is given by

(2.20) Q" = (--y"+ y"Eg"(xJ' , y")) dx"+((m- 1)x"+x"2tp"(x ", y")) dy".

If m - 1 = I, we see easily that blowing-up P" the exceptional divisor is a dicritical component. If m - 1 I>2, we reason by induction on m - 1 : blowing-up P" and looking at the point corresponding to (x"=0), then m - 1 decreases one unit. This is the desired contradiction.

Case 2: otherwise. Then the invariant p + q decreases strictly and we are done by

induction. []

The following proposition gives to us the formal normal forms for the non-dicritical pre-simple singularities in dimension two:

PROPOSITION 2.5. Let P be a pre-simple singularity for (,~, E). Then there are a regular system o f parameters (x,y) o f Ox.t" and a generator g2 o f @x such that

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SEPARATRICES AND DESINGULARIZATION OF FOLIATIONS 33 JE, P~xyOx, e and if2 is in one o f the following formal normal forms:

(i) ~=xy(dx/x+2dy/y); 2 E C , 2 ~ Q _ . (ii) f2=xyyS(dx/x +(e + 1/y ~) dy/y); s>- l, e E C.

(iii) ff2=xy(xPyq) s (dx/x +(e+ I/(xPyq) s) (p dx/x +q dy/y); g.c.d.(p, q)= 1, s~> 1.

(iv)* f~=x((my+ x m) dx/x-dy); m ~ l.

The formal normal forms (i), (ii) and (iii) correspond exactly to the simple singularities;

the formal normal form (iv)* corresponds to a pre-simple singularity which is not a simple singularity.

Conversely. Let (o ~, E)E ~(X, E), assume that P is the only point in Sing(Y, Q), where ((~, ~)=hol((~, E)). Assume that there is a regular system o f parameters (x, y) o f

~x,P satisfying

(a) 14=Je, e~xy~x.e;

(b) a generator Q o f ~p can be written down in either one o f the formal normal forms (i), (ii), (iii), or (iv)*;

(c) if(iv)*, then Je, e=X~x,e.

Then (~, E) is non-dicritical and P is a pre-simple singularity for (~, E).

Proof. Assume first that e(E, P ) = 1. Take a regular system of parameters (x, y) of 6x, e such that Je, e=X6x.e. Then ~e is generated by

(2.21) o)=a dx + b d y ; a, bEtTx, e, g . c . d . ( a , b ) = l ,

x

where ve(a)>~l (otherwise PC Sing(Y, ~)) and x does not divide a (otherwise E would be a dicritical component for (~, E)). Then ~e is generated by

(2.22) if2 = a dx + bx dy.

Since P is a pre-simple singularity, one of the following two possibilities holds:

(A) ve(b)=O, i.e. b is a unit of ~Tx, v.

(B) Ve (b)~ > 1 and Inl(a)=a Inl(x)+fl Inl(y) with f14:0.

Consider the possibility (A). Then b--l, up to multiply ~ by b -1. Thus ~ ( ~ , E ) p is generated by the germ of vector field

a a

(2.23) D = x - ~ - x - a ay"

By Proposition I. l, we have a regular system of parameters (x, y^) of ~x,e which is a linearizing system of parameters for Ds. Assume that Ds(Y^)=2y ^, for a certain 2 E C.

3-928285 Acta Mathematica 169. Imprim6 le 20 aorat 1992

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34 F. C A N O A N D D . C E R V E A U

(Note that 2 = - 0 a / 0 y ( 0 ) . ) Actually we have that

(2.24) I/2 = A(~, E; E; P).

Hence 2 ~ Q + - N 0 (l/N) by Lemma 2.4. Consider the following cases:

(AI) 2=0. Then, in view of (1.14), we can write D as follows:

c3 _ Z e J "(y^)j+l cO

(2.25) D = x Ox j>~l aY ^

Let s be the first index such that es*0. Note that s<oo, otherwise x divides the coefficients of g2. Hence, #e is generated over ~x,e by

where u(y ^) ~. t~x, e is a unit. A coordinate change y' = y ' ( y ^ ) allows us to write

( y

(2.27) u(Y^) .^,~.l - e +

( y )

for a certain residue e E C. Hence, multiplying g2 ^ by a unit in ~x,e, we have the normal form (ii).

(A2) 2 = - p / q E Q _ = - Q + ; with g.c.d.(p, q ) = l . We can take (2.28) D ₌ qx a _- _a cO _; _{- -}Oa ( 0 ) = p .

Ox ay ay

By (1.14), we have

0 a

(2.29) D = qX-~x - p y ^ Z eJ "(xpy^q)jy^ a .

OY ^ j>~l OY ^

Let s be the first index such that es*0. If s= co, we have the normal form (i). Assume that s < ~ . Then, ~p is generated by

x (xPy ^q)s P + --y^ ] '

where u(t)E C[[t]] is a unit. Take coordinate changes t ' = t ' ( t ) and y ' = y ' ( y ^ ) such that u(t) d_~t

(2.31) 9 = ~" ; xPy 'q= t'(xPy^q).

t s t \ t's] t'